Theorem funco 3625

Description: The composition of two functions is a function. Exercise 29 of [TakeutiZaring] p. 25.

Assertion

Ref	Expression
funco	|- ((Fun F /\ Fun G) -> Fun (F o. G))

Proof of Theorem funco

Step	Hyp	Ref	Expression
1		moexexv 1473	. . . . . . 7 |- ((E*z xGz /\ A.zE*y zFy) -> E*yE.z(xGz /\ zFy))
2		funmo 3607	. . . . . . 7 |- (Fun G -> E*z xGz)
3		dffun6 3606	. . . . . . . 8 |- (Fun F <-> (Rel F /\ A.zE*y zFy))
4	3	pm3.27bi 324	. . . . . . 7 |- (Fun F -> A.zE*y zFy)
5	1, 2, 4	syl2an 456	. . . . . 6 |- ((Fun G /\ Fun F) -> E*yE.z(xGz /\ zFy))
6	5	ancoms 438	. . . . 5 |- ((Fun F /\ Fun G) -> E*yE.z(xGz /\ zFy))
7		visset 1851	. . . . . . 7 |- x e. V
8		visset 1851	. . . . . . 7 |- y e. V
9	7, 8	brco 3352	. . . . . 6 |- (x(F o. G)y <-> E.z(xGz /\ zFy))
10	9	mobii 1438	. . . . 5 |- (E*y x(F o. G)y <-> E*yE.z(xGz /\ zFy))
11	6, 10	sylibr 198	. . . 4 |- ((Fun F /\ Fun G) -> E*y x(F o. G)y)
12	11	19.21aiv 1319	. . 3 |- ((Fun F /\ Fun G) -> A.xE*y x(F o. G)y)
13		relco 3567	. . 3 |- Rel (F o. G)
14	12, 13	jctil 290	. 2 |- ((Fun F /\ Fun G) -> (Rel (F o. G) /\ A.xE*y x(F o. G)y))
15		dffun6 3606	. 2 |- (Fun (F o. G) <-> (Rel (F o. G) /\ A.xE*y x(F o. G)y))
16	14, 15	sylibr 198	1 |- ((Fun F /\ Fun G) -> Fun (F o. G))

Syntax hints:   -> wi 3   /\ wa 221  A.wal 986  E.wex 1012  E*wmo 1414   class class class wbr 2669   o. ccom 3229  Rel wrel 3230  Fun wfun 3231

This theorem is referenced by:  fnco 3670  fco 3711  f1co 3742  fvco 3850  curry1 4208  curry2 4211  mapenlem1 4578  vsfval 8373

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 994  ax-gen 995  ax-8 996  ax-10 998  ax-11 999  ax-12 1000  ax-13 1001  ax-14 1002  ax-17 1003  ax-4 1005  ax-5o 1007  ax-6o 1010  ax-9o 1155  ax-10o 1173  ax-16 1243  ax-11o 1251  ax-ext 1494  ax-sep 2754  ax-pow 2794  ax-pr 2832

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1013  df-sb 1205  df-eu 1415  df-mo 1416  df-clab 1500  df-cleq 1505  df-clel 1508  df-ne 1624  df-v 1850  df-dif 2093  df-un 2094  df-in 2095  df-ss 2097  df-nul 2325  df-pw 2447  df-sn 2457  df-pr 2458  df-op 2461  df-br 2670  df-opab 2718  df-id 2889  df-xp 3239  df-rel 3240  df-cnv 3241  df-co 3242  df-fun 3247

Copyright terms: Public domain